The degrees of a system of parameters of the ring of invariants of a binary form

We consider the degrees of the elements of a homogeneous system of parameters for the ring of invariants of a binary form, give a divisibility condition, and a complete classification for forms of degree at most 8. 1 The degrees of a system of parameters Let R be a graded C-algebra. A homogeneous system of parameters (hsop) of R is an algebraically independent set S of homogeneous elements of R such that R is module-finite over the subalgebra generated by S. By the Noether normalization lemma, a hsop always exists. The size |S| of S equals the Krull dimension of R. In this note we consider the special case where R is the ring I of invariants of binary forms of degree n under the action of SL(2,C). This ring is CohenMacaulay, that is, I is free over the subring generated by any hsop S. Its Krull dimension is n− 2. One cannot expect to classify all hsops of I. Indeed, any generic subset with the right degrees will be a hsop (cf. Dixmier’s criterion below). But one can expect to classify the sets of degrees of hsops. In this note we give a divisibility restriction on the set of degrees for the elements of a hsop, and conjecture that when all degrees are large this restriction also suffices for the existence of a hsop with these given degrees. For small degrees there are further restrictions. We give a complete classification for n ≤ 8. 2 Hilbert’s criterion Hilbert’s criterion gives a characterization of homogeneous systems of parameters as sets that define the nullcone. Denote by Vn the set of binary forms of degree n. The nullcone of Vn, denoted N (Vn), is the set of binary forms of degree n on which all invariants vanish. By the Hilbert-Mumford numerical criterion (see [6] and [7, Chapter 2]) this is precisely the set of binary forms of degree n with a root of multiplicity > n2 . Moreover, the binary forms with no root of multiplicity ≥ n 2 have closed SL(2,C)-orbits. The elements of N (Vn) are called nullforms. Another result from [6] that we will use is the following.